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%I #52 Jan 21 2024 15:29:44
%S 1,9,45,55,99,297,703,999,2223,2728,4950,5050,7272,7777,9999,17344,
%T 22222,77778,82656,95121,99999,142857,148149,181819,187110,208495,
%U 318682,329967,351352,356643,390313,461539,466830,499500,500500,533170,538461,609687,643357
%N Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.
%C Consider an m-digit number n. Square it and add the right m digits to the left m or m-1 digits. If the resultant sum is n, then n is a term of the sequence.
%C 4879 and 5292 are in A006886 but not in this version.
%C Shape of plot (see links) seems to consist of line segments whose lengths along the x-axis depend on the number of unitary divisors of 10^m-1 which is equal to 2^w if m is a multiple of 3 or 2^(w+1) otherwise, where w is the number of distinct prime factors of the repunit of length m (A095370). w for m = 60 is 20, whereas w <= 15 for m < 60. This leads to the long segment corresponding to m = 60. - _Chai Wah Wu_, Jun 02 2016
%D D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.
%H Chai Wah Wu, <a href="/A053816/b053816.txt">Table of n, a(n) for n = 1..50000</a>
%H D. E. Iannucci, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/iann2a.html">The Kaprekar numbers</a>, J. Integer Sequences, Vol. 3, 2000, #1.2.
%H R. Munafo, <a href="http://www.mrob.com/pub/math/seq-kaprekar.html">Kaprekar Sequences</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KaprekarNumber.html">Kaprekar Number</a>
%H Chai Wah Wu, <a href="/A053816/a053816.pdf">Semilog plot of A053816(n), n = 1..1003371 (all terms with m <= 60).</a>
%H Chai Wah Wu, <a href="/A053816/a053816_1.pdf">Plot of A053816(n), n = 1..1003371 (all terms with m <= 60).</a>
%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%e 703 is Kaprekar because 703=494+209, 703^2=494209.
%t kapQ[n_]:=Module[{idn2=IntegerDigits[n^2],len},len=Length[idn2];FromDigits[ Take[idn2,Floor[len/2]]]+FromDigits[Take[idn2, -Ceiling[len/2]]]==n]; Select[Range[540000],kapQ] (* _Harvey P. Dale_, Aug 22 2011 *)
%t ktQ[n_] := ((x = n^2) - (z = FromDigits[Take[IntegerDigits[x], y = -IntegerLength[n]]]))*10^y + z == n; Select[Range[540000], ktQ] (* _Jayanta Basu_, Aug 04 2013 *)
%t Select[Range[540000],Total[FromDigits/@TakeDrop[IntegerDigits[#^2], Floor[ IntegerLength[ #^2]/2]]] ==#&] (* The program uses the TakeDrop function from Mathematica version 10 *) (* _Harvey P. Dale_, Jun 03 2016 *)
%o (Haskell)
%o a053816 n = a053816_list !! (n-1)
%o a053816_list = 1 : filter f [4..] where
%o f x = length us - length vs <= 1 &&
%o read (reverse us) + read (reverse vs) == x
%o where (us, vs) = splitAt (length $ show x) (reverse $ show (x^2))
%o -- _Reinhard Zumkeller_, Oct 04 2014
%Y Cf. A006886, A037042, A053394, A053395, A053396, A053397, A045913, A003052, A055642, A095370.
%K nonn,nice,base,easy
%O 1,2
%A _Robert Munafo_
%E More terms from _Michel ten Voorde_, Apr 11 2001
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